Nonexistence of exact solutions agreeing with the Gaussian beam on the beam axis or in the focal plane

Solutions of the Helmholtz equation which describe electromagnetic beams (and also acoustic or particle beams) are discussed. We show that an exact solution which reproduces the Gaussian beam waveform on the beam axis does not exist. This is surprising, since the Gaussian beam is a solution of the paraxial equation, and thus supposedly accurate on and near the beam axis. Likewise, a solution of the Helmholtz equation which exactly reproduces the Gaussian beam in the focal plane does not exist. We show that the last statement also holds for Bessel-Gauss beams. However, solutions of the Helmholtz equation (one of which is discussed in detail) can approximate the Gaussian waveform within the central focal region.